Law of Cosines

Definition

The Law of Cosines is a fundamental formula used in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. This formula is especially useful for solving triangles that are not right triangles, allowing the calculation of unknown side lengths or angle measures based on known values. It serves as a crucial tool for connecting side lengths and angles in various types of triangles, expanding the application of trigonometric principles beyond just right triangles.

5 Must Know Facts For Your Next Test

1. The Law of Cosines can be expressed as: $$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$, where a, b, and c are the sides of the triangle and C is the angle opposite side c.
2. It can also be rearranged to find angles when all sides are known, making it versatile for different types of problems.
3. This law generalizes the Pythagorean Theorem; when angle C is 90 degrees, the formula simplifies to that theorem.
4. It's particularly useful in navigation and physics where determining distances and angles is crucial.
5. Understanding how to apply this law allows for solving complex problems involving non-right triangles in various real-world contexts.

Review Questions

• How does the Law of Cosines differ from the Pythagorean Theorem, and when is it specifically used?
  • The Law of Cosines differs from the Pythagorean Theorem as it applies to any triangle, not just right triangles. While the Pythagorean Theorem specifically relates to right triangles where one angle is 90 degrees, the Law of Cosines can be utilized to find unknown side lengths or angle measures in oblique triangles. This makes it particularly useful in situations where only non-right triangles are involved.
• Illustrate how you would use the Law of Cosines to find an unknown side length in a triangle given two sides and the included angle.
  • To find an unknown side length using the Law of Cosines, first identify your known values: two side lengths (a and b) and the included angle (C). Plug these values into the formula: $$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$. Solve for c by calculating $$c = \sqrt{a^2 + b^2 - 2ab \cdot \cos(C)}$$, providing you with the length of the unknown side.
• Evaluate a scenario where the Law of Cosines would be necessary for solving a problem involving triangular distances in real life. What steps would you take?
  • In a scenario like triangulating locations on a map where two points are known and the angle between them is given, using the Law of Cosines becomes essential. First, measure the known distances between two points and determine the angle formed at one endpoint. Using these values, apply the Law of Cosines to calculate the distance from this endpoint to a third point. This process allows accurate determination of unknown distances, which is vital for navigation and geographical mapping.

Related terms

Pythagorean Theorem: A principle in geometry that states in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Sine Rule: A formula used to relate the lengths of the sides of a triangle to the sines of its angles, useful for non-right triangles.

Angle Measure: The amount of rotation needed to bring one side of an angle into alignment with another side, usually measured in degrees or radians.